I have created 2 prototypes which have been fairly successful in identifying the feel of the game I want, and both of these have different animation techniques.

The one in question is the skeletal animation. My problem is this: I don't have depth.

Perhaps the player would swing his sword, causing foreshortening in both his arms and sword. With a sprite, I cant show depth like that. How would this be handled?

The only way I could think of is replacing the mesh when necessary. Is there another way? The reason I don't want to use sprites is because they aren't very flexible, and 3D modeling takes a lot of skill which I don't have, and a lot of time. If this problem has been tackled in another game, please do point me to it, as I would also like to see how it looks in a finished product.

I need a little more clarification...
Is this going to be a 2D game with sprites?
How is the sword swinging animation going to look? Is it going to be a sideways slash, or over-the-head smash?
Are you using an existing game engine? If so, what are you using?
Have you thought about purchasing existing art assets and animations?

Generally, doing 2d skeletal animation...you won't be able to get the changes you are looking for without changing the image. You might be able to kind of fake it by doing some scaling or something similar, but I don't know how easy to get the right look it would be. This is in fact the primary reason why the animation software like Spine and Spriter let you change images as part of the animation.

Ok well this honestly does seem like it would be finicky or aesthetically unpleasing. I wanted to know if there was some way that I haven't been exposed to for some reason to solve the problem. Perhaps the extra work for sprite based animation is worth it.